Keller's box-scheme for the one-dimensional stationary convection-diffusion equation

The box-scheme of H. B. Keller. initially derived in [22] for the one-dimensional heat equation. is a mixed finite volume scheme for conservative equations. The basic principle of the scheme for equations like div phi(u.Deltau) = f. is to take the average onto the same mesh of the two equations of the mixed form. the conservation law div = f and the constitutive law p = phi(u.Deltau). In this Paper we perform the numerical analysis of two Keller-like box-schemes for the one-dimensional convection-diffusion equation cu(chi) - epsilonu(chichi) = f. In the first one, introduced by B. Courbet in [9.10], the numerical average of the diffusive flux is upwinded along the sign of the velocity, giving a first order accurate scheme. The second one is fourth order accurate, It is baked onto the Euler-MacLaurin quadrature formula for the average of the diffusive flux . We emphasize in each case the link with the SUPG finite element methods